# PDF superior TítuloA residual based a posteriori error estimator for an augmented mixed finite element method in linear elasticity ### TítuloA residual based a posteriori error estimator for an augmented mixed finite element method in linear elasticity

In Tables 5.5 through 5.10 we provide the individual and total errors, the experimental rates of convergence, the a posteriori error estimators, and the eﬀectivity indices for the uniform and adaptive reﬁnements as applied to Examples 3–5. In this case, uniform reﬁnement means that, given a uniform initial triangulation, each subsequent mesh is obtained from the previous one by dividing each triangle into the four ones arising when connecting the midpoints of its sides. We observe from these tables that the errors of the adaptive procedure decrease much faster than those obtained by the uniform one, which is conﬁrmed by the experimental rates of convergence provided there. This fact can also be seen in Figures 5.1 through 5.3 where we display the total error e (σ, u , γ) vs. the degrees of freedom N for both reﬁnements. As shown by the values of r ( e ), particularly in Example 3 (where r ( e ) approaches 1 / 3 for the uniform reﬁnement), the adaptive method is able to recover, at least approximately, the quasi-optimal rate of convergence O ( h ) for the total error. Furthermore, the eﬀectivity indices remain again bounded from above and below, which conﬁrms the reliability and eﬃciency of θ for the adaptive algorithm. On the other hand, some intermediate meshes obtained with the adaptive reﬁnement are displayed in Figures 5.4 through 5.6. Note that the method is able to recognize the singularities and the large stress regions of the solutions. In particular, this fact is observed in Example 3 (see Fig. 5.4) where the adapted meshes are highly reﬁned around the singular point (0 , 0). Similarly, the adapted meshes obtained in Examples 4 and 5 (see Figs. 5.5 and 5.6) concentrate the reﬁnements around the interior point (1 / 2 , 1 / 2) and the segment x 1 = 0, respectively, where the largest stresses occur.
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27 Lee mas ### TítuloLow cost a posteriori error estimators for an augmented mixed FEM in linear elasticity

We remark that the a posteriori error estimator ¯ θ is locally efficient in the interior elements (that is, those that do not touch the boundary). Besides, the computation of ¯ θ involves four residuals per element in the interior tri- angles, five residuals per element in the triangles with exactly one vertex on the boundary and six residuals per element in the triangles with a side on the boundary. We emphasize that ¯ θ has been derived in the two-dimensional set- ting, assuming that displacements are approximated by continuous piecewise linear finite elements. In the next section, we provide some numerical results illustrating the performance of the corresponding adaptive method, including a numerical comparison with the a posteriori error estimator proposed in . 4. Numerical experiments
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38 Lee mas ### TítuloA mixed finite element method for the generalized Stokes problem

We present and analyse a new mixed finite element method for the generalized Stokes prob- lem. The approach, which is a natural extension of a previous procedure applied to quasi- Newtonian Stokes flows, is based on the introduction of the flux and the tensor gradient of the velocity as further unknowns. This yields a two-fold saddle point operator equation as the resulting variational formulation. Then, applying a slight generalization of the well known Babuˇ ska-Brezzi theory, we prove that the continuous and discrete formulations are well posed, and derive the associated a priori error analysis. In particular, the finite element subspaces providing stability coincide with those employed for the usual Stokes flows except for one of them that needs to be suitably enriched. We also develop an a-posteriori error estimate (based on local problems) and propose the associated adaptive algorithm to com- pute the finite element solutions. Several numerical results illustrate the performance of the method and its capability to localize boundary layers, inner layers, and singularities.
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27 Lee mas ### Primal and mixed finite element methods for image registration

Deformable Image Registration (DIR) is a powerful computational method for image analysis, with promising applications in the diagnosis of human disease. Despite being widely used in the medical imaging community, the mathematical and numerical analysis of DIR methods still has many open questions. Further, recent applications of DIR in- clude the quantification of mechanical quantities in addition to the aligning transformation, which justifies the development of novel DIR formulations for which the accuracy and convergence of fields other than the aligning transformation can be studied. In this work we propose and analyze a primal, mixed and augmented formulations for the DIR prob- lem, together with their finite-element discretization schemes for their numerical solution. The DIR variational problem is equivalent to the linear elasticity problem with a source term that has a nonlinear dependence on the unknown field. Fixed point arguments and small data assumptions are employed to derive the well-posedness of both the continuous and discrete schemes for the usual primal and mixed variational formulations, as well as for an augmented version of the later. In particular, continuous piecewise linear elements for the displacement in the case of the primal method, and Brezzi-Douglas-Marini of order 1 (resp. Raviart-Thomas of order 0) for the stress together with piecewise constants (resp. continuous piecewise linear) for the displacement when using the mixed approach (resp. its augmented version), constitute feasible choices that guarantee the stability of the asso- ciated Galerkin systems. A priori error estimates derived by using Strang-type lemmas, and their associated rates of convergence depending on the corresponding approximation properties are also provided. Numerical convergence tests and DIR examples are included to demonstrate the applicability of the method.
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71 Lee mas ### TítuloA low order mixed finite element method for a class of quasi Newtonian Stokes flows Part II: a posteriori error analysis

On the other hand, we recall that the application of adaptive algorithms, based on a posteriori error estimates, usually guarantees the quasi-optimal rate of convergence of the ﬁnite element solution to boundary value problems. In addition, this adaptivity is specially necessary for nonlinear problems where no a priori hints on how to build suitable meshes are available. To this respect, we have shown recently that the combination of the usual Bank–Weiser approach from  with the analysis from [3,4] allows to derive fully explicit and reliable a posteriori error estimates for the dual-mixed variational formulations (showing a two- fold saddle point structure) of some linear and nonlinear problems (see, e.g. [2,6,7]). However, no a pos- teriori error analysis has been developed yet for the nonlinear Stokes problems studied in . Therefore, as a natural continuation of our results in , in the present paper we apply the Bank–Weiser type a posteriori error analysis mentioned above to derive reliable estimates for the mixed ﬁnite element scheme (1.6). The rest of this work is organized as follows. In Section 2 we collect some basic results on Sobolev spaces and state the main result of this paper. The proof of our a posteriori estimate, which makes use of the Ritz projection of the error, is provided in Section 3. In Section 4 we prove the quasi-eﬃciency of the estimator and discuss on suitable choices for the auxiliary functions needed for its computation. Finally, several numerical results illustrating the good performance of the adaptive algorithm are reported in Section 5.
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19 Lee mas ### Primal and Mixed Finite Element Methods for Deformable Image Registration Problems

Deformable image registration (DIR) represent a powerful computational method for image analy- sis, with promising applications in the diagnosis of human disease. Despite being widely used in the medical imaging community, the mathematical and numerical analysis of DIR methods remain un- derstudied. Further, recent applications of DIR include the quantification of mechanical quantities apart from the aligning transformation, which justifies the development of novel DIR formulations where the accuracy and convergence of fields other than the aligning transformation can be stud- ied. In this work we propose and analyze a primal, mixed and augmented formulations for the DIR problem, together with their finite-element discretization schemes for their numerical solution. The DIR variational problem is equivalent to the linear elasticity problem with a nonlinear source term that depends on the unknown field. Fixed point arguments and small data assumptions are em- ployed to derive the well-posedness of both the continuous and discrete schemes for the usual primal and mixed variational formulations, as well as for an augmented version of the later. In particular, continuous piecewise linear elements for the displacement in the case of the primal method, and Brezzi-Douglas-Marini of order 1 (resp. Raviart-Thomas of order 0) for the stress together with piecewise constants (resp. continuous piecewise linear) for the displacement when using the mixed approach (resp. its augmented version), constitute feasible choices that guarantee the stability of the associated Galerkin systems. A-priori error estimates derived by using Strang-type Lemmas, and their associated rates of convergence depending on the corresponding approximation properties are also provided. Numerical convergence tests and DIR examples are included to demonstrate the applicability of the method.
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27 Lee mas ### TítuloStabilized dual mixed method for the problem of linear elasticity with mixed boundary conditions

We extend the applicability of the augmented dual-mixed method introduced recently in [4, 5] to the problem of linear elasticity with mixed boundary conditions. The method is based on the Hellinger–Reissner principle and the symmetry of the stress tensor is imposed in a weak sense. The Neuman boundary condition is prescribed in the finite element space. Then, suitable Galerkin least-squares type terms are added in order to obtain an augmented variational formulation which is coercive in the whole space. This allows to use any finite element subspaces to approximate the displacement, the Cauchy stress tensor and the rotation.
Mostrar más ### TítuloA low order mixed finite element method for a class of quasi Newtonian Stokes flows Part I: a priori error analysis

In the recent papers [3,16] we analyzed dual-mixed formulations for non-linear boundary value problems in plane elasticity. In the case of incompressible materials, we considered the non-Newtonian model from [5,7], and applied the dual-mixed approach from  to study its solvability and ﬁnite element approxi- mations. Since the non-linear constitutive law depends on the strain tensor, we introduced this variable and the rotation as further unknowns, which yielded a twofold saddle point operator equation as the resulting variational formulation. Then, we extended the well known PEERS space and deﬁned a stable Galerkin scheme, for which a Bank–Weiser type a posteriori error analysis was also developed.
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12 Lee mas ### TítuloA posteriori error analysis of an augmented mixed finite element method for Darcy flow

We develop an a posteriori error analysis of residual type of a stabilized mixed finite element method for Darcy flow. The stabilized formulation is obtained by adding to the standard dual- mixed approach suitable residual type terms arising from Darcy’s law and the mass conservation equation. We derive sufficient conditions on the stabilization parameters that guarantee that the augmented variational formulation and the corresponding Galerkin scheme are well-posed. Then, we obtain a simple a posteriori error estimator and prove that it is reliable and locally efficient. Finally, we provide several numerical experiments that illustrate the theoretical results and support the use of the corresponding adaptive algorithm in practice.
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20 Lee mas ### TítuloA posteriori error analysis of an augmented mixed formulation in linear elasticity with mixed and Dirichlet boundary conditions

The approach there is based on the introduction of suitable Galerkin least-squares terms that arise from the constitutive and equilibrium equations, and from the relation defining the rotation in terms of the displacement. In , besides these Galerkin least- squares terms, a consistency term related with the non-homogeneous Dirichlet boundary condition is added. In the case of pure Dirichlet boundary conditions, the bilinear form of the augmented formulation is bounded and coercive on the whole space and hence, the associated Galerkin scheme is well-posed for any finite element subspace. Thus, it is possible to use as finite element subspaces some non-feasible choices for the usual (non-augmented) dual-mixed formulation. In particular, it is possible to employ Raviart- Thomas elements of lowest order to approximate the stress tensor, continuous piecewise linear elements for the displacement, and piecewise constants for the rotation. In the case of mixed boundary conditions, the trace of the displacement on the Neumann boundary can be approximated by continuous piecewise linear elements on an independent partition of that boundary whose mesh size needs to satisfy a compatibility condition with the mesh size of the triangulation of the domain.
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31 Lee mas ### ADAPTIVE SPACE-TIME FINITE ELEMENT METHODS FOR ACOUSTICS SIMULATIONS IN UNBOUNDED DOMAINS

The development of the space-time method proceeds by considering a partition M of the total time interval, t ∈ J = (0, T ), 0 < T < ∞ , into N time steps { I n } N n=0 given by I n = { (t n , t n+1 ) } N n=0 . The length of the variable time step is given by ∆t n = t n+1 − t n . Using this notation, Q n = Ω × I n , are the nth space-time slabs. Within each space-time slab, the spatial domain is subdivided into D n elements. We define the jump operator across space-time slabs as, ### TítuloA stabilized finite element approach for advective diffusive transport problems

Almeida (Editors); Escola Politécnica da Universidade de Sao Paulo, Sao Paulo, Brasil.[r]

20 Lee mas ### Grad div stabilization for the time dependent Boussinesq equations with inf sup stable finite elements

the velocity and temperature equations and LPS grad-div stabilization. In  the weak convergence of the method is obtained but no error bounds for the method are proved. In particular, since no bounds are proved for the rate of convergence of the method the question of the dependence on the constants in the bounds on the Rayleigh numbers is not considered. Our aim in the present paper is, on the one hand, proving error bounds for the methods with constants that do not deteriorate for increasing values of the Rayleigh numbers and, on the other hand, doing this with the fewest possible stabilization terms. As a consequence, error bounds are obtained with constants independent on inverse powers of ν and κ in formulation (2) with only grad-div stabilization.
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17 Lee mas ### An application of the finite element method to curve fitting

From the two above examples it can be deducted that the FEM allows to define unequivocally the curve of the road axis, subject to be continuous c2 and minimizing a functional of the type (1). In this way, it is not ne cessary to use the traditional special curves such as straight lines, circles and clothoides, and therefore a more wide freedom for the road designer is reached. The axis can be defined using the FEM simply by the n £ de coordinates of some special points (joints) and so- me extra constraints in the slope and/or curvature. The coordinates of intermediate points between two con secutive joints can be obtained from the use of the shape or interpolation functions (6).
Mostrar más ### Two-dimensional mesh optimization in the finite element method

Abstract-The solution to the problem of finding the optimum mesh design in the finite element method with the restriction of a given number of degrees of freedom, is [r] ### A Fixed Grid method for hyperelastic models in large strain analysis using the level set method

This system of equations are solved by use of an incremental formulation. Here ˆ u and ˆ p are the vectors of nodal point incremental displacements and pressures, respectively. t+∆t R is the vector of externally applied nodal point loads corresponding to time t + ∆t; and F = t FU, t FP T is the vector of nodal point forces corresponding to the internal element stresses at time t. Due to this, the subtraction at the right hand side is known as the out-of-balance force vector and may reach an small value at the end of each load increment. Generally, this value is established as a percentage of the first out-of-balance vector magnitude at a load increment.
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12 Lee mas ### Value added in hierarchical linear mixed model with error in variables

In this thesis, we propose a novel method to address the problem of Value Added estimation in Hierarchical Linear Mixed Models with measurement error in the variables. Unlike the other proposals, we begin from a general two level model with K explanatory variables measured with error, a model of measurement error and a specific formula for Value Added for a school as suggested by Manzi et.al (2014) . Also, it is important to mention that our proposal considers a general model instead of a particular model and additionally we include the specific expression for Value Added which was not taken into account by previous authors. Moreover, we consider the measurement error based on the specification of the model reaching the stage of the estimation of the parameters and thus our new method not only permits estimating the fixed parameters and variances of the residual error and the random effect, but also the Value Added of the schools. The Value Added estimator is obtained using Empirical Bayes estimators and under the new specification of the model given in our proposal. Another important contribution is the adjustment of a resampling method to our method for obtaining an uncertainty measurement for the Value Added estimator.
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89 Lee mas ### An ADER finite volume method for an atherosclerosis model

•We have obtained a numerical solution of a 1D nonlinear reaction-diffusion type model, representing the initial stages of atherosclerosis, which is a variant of the one prop[r]

71 Lee mas ### A English

The use of a software based on the finite element method, is a good support for the realization of engineering designs that involve the determination of stress, displacement and strain in complex parts, especially with stress concentrators. Also, the software allows to determine the distribution of tensions in zones with concentrators of stresses with geometric form not defined analytically. ### TítuloA 2D numerical model using finite volume method for sediment transport in rivers

It is very frequent in scientific literature to use the dam-break problem for the validation of a model. A domain of 5x200 m without friction at the bottom has been considered, ensuring depths of 1 m and 0.1 m as initial conditions in the two separated areas. Again the water depth in the axis of symmetry, at time t = 25 s, is presented. The ∆t used (0.2 s) is the maximum one in which there are no oscillations, so is equivalent to Courant´s number for one dimen- sion. Results match very well the exact solution as can be seen in Burguete & García-Navarro (2000).
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